Chaotic dynamic physical systems are dynamic physical systems that exhibit behaviour that is deterministic in a mathematical sense (the behaviour is precisely determined by the state at any particular time), but nevertheless unpredictable over time due to the sensitivity of the system on the state. Due to this unpredictability, when attempting to control a dynamic physical system it is generally considered undesirable to allow it to enter a chaotic state, as this limits the ability to control the system.
However, avoiding chaotic states may mean restricting the system to inefficient states. For example, in a chemical reaction, while they are unpredictable the chaotic states may make use of resources more efficiently than the stable, non-chaotic states. Further, in systems beyond a certain size chaotic behaviour may be inevitable, and so chaotic behaviour can be avoided only by restricting to very small-scale systems. For this reason, the control of chaotic systems has been of some interest.
A dynamic physical system can be represented by a plurality of variables, each representing a quantity of the system that varies over time. (For example, if the system is a chemical process, the variables could represent the quantities of the different chemicals or their concentrations.) The system can then be modelled by a plurality of rate equations for each of the variables, which describe how the quantities vary over time based on the current state.
As is well known, the instantaneous state of a system can be considered to be a point moving around a “state space”, where the dimensions of the state space represent the different variables of the system, and the position of the point at a given time is determined by the values of the variables at that time. The state of an example Rössler system (a well-known simple dynamic physical system that exhibits chaotic behaviour) as a point moving through state space is shown in FIG. 1.
It is known that for dynamic physical systems in ranges where chaotic behaviour occurs, the global behaviour of the system is dominated by a subset of the rate equations, which tend to be the non-linear parts of the system that allow the quantities of the system to grow at an exponential rate. The local rate of expansion is proportional to the local behaviour of each of the variables.
A known method of controlling a chaotic system is the OGY (Ott, Grebogi and Yorke) method, as first described in E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos”, Phys. Rev. Lett. 64 1196 (1990). The OGY model is based upon the properties of chaotic systems discussed above. In the OGY method, a model of the dynamic physical system is obtained, and analysed to identify an unstable periodic orbit around which the system cycles is identified. The system is then controlled by applying small, pre-determined changes to the variables of the system, in proportion to the local behaviour of the variables, in order to keep the system in or near the chosen orbit.
Another known method of controlling a chaotic system is the Pyragas continuous control method. Similarly to the OGY method, an unstable periodic orbit around which the system cycles is identified. However, in this method adjustments are made to the system in accordance with pre-determined time delays, which need to be carefully matched with the dynamics of the system to allow successful control.
It is a disadvantage of the both OGY method and the Pyragas method that they require the system in operation to be analysed in order to identify a particular unstable periodic orbit, so that the properties of that orbit can be used to implement the method. Further, neither method allows control of systems in all cases.
The present invention seeks to mitigate the above-mentioned problems. Alternatively or additionally, the present invention seeks to provide an improved method of controlling a dynamic physical system, in particular a dynamic physical system that exhibits chaotic behaviour.